Plane Geometry - Bruce E. Shapiro

SECTION 49. TRIANGLES IN HYPERBOLIC GEOMETRY 287Figure 49.4: Continuity of Defect.Proof. Let ɛ > 0 be given and suppose that x ∈ [0, c].By theorem 49.4 there is a d > 0 such then whenever P (x)P (y) thenδ(△CP (x)P (y)) < ɛ (see figure 49.4).Choose y ∈ [0, c] such that |x − y| < d.Then P (x)P (y) = |x − y| < d which implies that δ(CP (x)P (y)) < ɛ.By the additivity of defect,|f(x) − f(y)| = |δ(△AP (x)C) − δ(△AP (y)C)|= δ(△CP (x)P (y))< ɛTheorem 49.6 The defect is onto. Specifically, ∀y ∈ (0, 180) there existsa triangle △ABC such that δ(△ABC) = y.Proof. Let y ∈ (0, 180) be given.Let ɛ = 180 − y.Since 0 < y < 180, then 0 < ɛ < 180.By theorem 49.1, there exists a triangle △ABC such thatδ(△ABC) > 180 − ɛ = yBy theorem 49.2, there exists a triangle △A ′ B ′ C ′ such thatδ(△A ′ B ′ C ′ ) < yBy continuity of defect and the intermediate value theorem, there existssome triangle with defect y.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.